From invasion to coexistence: A mathematical modeling approach to predator–prey dynamics under invasive pressure

Biological invasion stands out as one of the most crucial ecological phenomena within an ecosystem. Invasion indicates the phenomenon in which a novel species infiltrates a pre-existing ecosystem. Invasive predator species present considerable challenges to native ecosystems, frequently altering established predator–prey relationships. Driven by empirical observations and a conspicuous lack of comprehensive mathematical modeling in the context of ecological invasions, this study formulates and examines a mathematical model that represents the ecological dynamics in a post-invasion context, featuring two predator species – one native and one invasive – alongside a common prey species. The model integrates essential ecological processes, including the utilization of prey refuges to evade native predators, prey naivety toward invasive predators, the counter-attack strategies employed by prey, interspecific competition among predators, the reintroduction or migration of native predators, environmental challenges impacting invasive predators, and the harvesting efforts aimed at controlling the invasive species. We have conducted rigorous stability and bifurcation analyses, including two-parametric bifurcation analysis, to gain insights into the system dynamics. Detailed numerical simulations are conducted to explore the impacts of reintroducing or migrating native predators and various other parameters on the long-term effects of invasion, including conditions for species coexistence, invasion failure, and the extinction of native predators. The results of our study yield fresh theoretical perspectives on the processes that influence post-invasion dynamics, while also suggesting possible approaches for managing the effects of invasive predators on native ecosystems. • Developed a three-species model involving prey, native predator, and invasive predator to study post-invasion dynamics. • Incorporated ecological traits like prey refuge, counter-attack, and naivety toward invasive predators. • Modeled reintroduction of native predators, interspecific competition, and invasive predator harvesting. • Multiple invasion outcomes observed: success, failure, or stable coexistence. • Native predator reintroduction or migration rate critically influences system stability and invasion outcome.